Vapor compression systems, such as heat pumps, refrigeration and air-conditioning systems, are widely used in industrial and residential applications. The introduction of variable speed compressors, variable position valves, and variable speed fans for the vapor compression cycle has greatly improved the flexibility of the operation of such systems. It is possible to improve the efficiency of vapor compression systems by controlling these new components correctly.
For example, a speed of the compressor can be adjusted to modulate a flow rate of a refrigerant. The speed of an evaporator fan and a condenser fan can be varied to alter heat transfer between air and heat exchangers. The change in an expansion valve opening can directly influence a pressure drop between a high-pressure and a low-pressure in the vapor compression system, which, in turn, affects the flow rate of the refrigerant as well as superheat temperature at the corresponding evaporator outlet.
A combination of commanded inputs to the vapor compression system that delivers a particular amount of heat is often not unique and these various combinations consume different amounts of energy. Therefore, it is desirable to operate the vapor compression system using the combination of inputs that minimizes energy and maximizes efficiency of the system.
Conventionally, methods maximizing the energy efficiency rely on the use of mathematical models of the physics of vapor compression systems. Those model-based methods attempt to describe the influence of commanded inputs of the components of the vapor compression system on the thermodynamic behavior of the system and the consumed energy. In those methods, models are used to predict the combination of inputs that both meets the heat load requirements and minimizes energy.
However, the use of mathematical models for the selection of optimizing inputs has several important drawbacks. Firstly, models rely on simplifying assumptions in order to produce a mathematically tractable representation. These assumptions often ignore important effects or do not consider installation-specific characteristics such as room size, causing the model of the system to deviate from actual behavior of the system.
Secondly, the variation in those systems during the manufacturing process are often so large as to produce vapor compression systems of the same type that exhibit different input-output characteristics, and therefore a single model cannot accurately describe the variations among copies produced as the outcome of a manufacturing process.
Thirdly, those models are difficult to derive and calibrate. For example, parameters that describe the operation of a component of a vapor compression system, e.g., a compressor, are experimentally determined for each type of the compressor used, and a model of a complete vapor compression system may have dozens of such parameters. Determining the values of these parameters for each model is an extensive effort. Also, vapor compression systems are known to vary over time. A model that accurately describes the operation of a vapor compression system at one time may not be accurate at a later time as the system changes, for example, due to slowly leaking refrigerant, or the accumulation of corrosion on the heat exchangers.
An alternative to the model-based controllers optimizing a metric of performance includes “extremum-seeking” controllers (ESC) due to their ability to maximize or minimize a signal of interest. Conventional ESC actively experiment with the device under control by applying a perturbation to one or more inputs, and measuring the resulting perturbations in the performance metric. These perturbations are averaged over some time window to produce an estimate of the gradient of the performance metric. This local estimate of the gradient is then used to steer the average value of the inputs in the direction of the gradient that maximizes or minimizes the cost.
For example, the method described in U.S. Pat. No. 8,694,131 teaches that a perturbation-based extremum seeking controller can be configured to modify the operation of a vapor compression system such that energy-optimal combinations of actuators are used to direct the operation of vapor compression systems. While the perturbation-based extremum seeking method can achieve the optimum of a convex performance metric without relying on a model, that method suffers from slow convergence rates. Because the objective of ESC is to find an optimal steady state operating point, the extremum seeking controller controls the plant in a quasi-steady manner, i.e., without exciting the plant's dynamic response. Otherwise, phase information between the applied controls and measurements due to the transient response cannot be distinguished from the phase information due to the sinusoidally perturbed measurement of the performance metric.
If the slowest, and therefore dominant, time constant associated with the natural dynamics of the vapor compression system is called τplant, then the perturbation period τperturb must be much slower (larger time constant): τperturb>>τplant. Further, the ESC must average several perturbations in order to obtain an accurate estimate of the (average) gradient, and since the extremum seeking occurs on the timescale of this averaged gradient, the convergence rate of the extremum seeking controller is two time scales slower than the plant dynamics:τadapt>>τperturb>>τplant 
Because the dominant time constant of the vapor compression system is often on the order of tens of minutes, the extremum seeking controller can take several hours to converge to the optimum point. And since the disturbances acting on the vapor compression system are known to have faster dynamics, the optimal operating point can change before the perturbation ESC converges. As a result, the slow convergence property of perturbation-based extremum seeking represents a barrier to the solution of real time optimization of the performance of vapor compression systems.
Researchers have struggled with the slow convergence rates of perturbation-based ESC for some time. Initial efforts focused on introducing filters that separated the effect of the phase of the transient part of the response from the response to the perturbation. However, that method requires that the filters be designed with specific and detailed knowledge of the plant, and even when this information is available, the convergence rate is only marginally improved because the perturbation averaging is still required.
Others method have considered estimating the gradient of the performance metric, and a Hessian matrix. However, because the value of the Hessian matrix quickly approaches zero as the optimum point is approached, that method quickly becomes overwhelmed with noise.
Accordingly, there is a need in the art to improve the rate of convergence of the extremum seeking controllers.